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%I #38 Oct 30 2023 02:00:27
%S 0,1,258,6564,66052,390630,1693512,5764808,16909320,43066413,
%T 100782540,214358892,433565328,815730734,1487320464,2564095320,
%U 4328785936,6975757458,11111134554,16983563060,25801892760,37840199712,55304594136,78310985304,110992776480
%N Coefficients in q-expansion of E_4*(E_2*E_4 - E_6)/720, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.
%C Multiplicative because A013955 is. - _Andrew Howroyd_, Jul 25 2018
%H Seiichi Manyama, <a href="/A282060/b282060.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f.: phi_{8, 1}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.
%F a(n) = (A282101(n) - A013974(n))/720. - _Seiichi Manyama_, Feb 10 2017
%F If p is a prime, a(p) = p^8 + p = A196288(p). - _Seiichi Manyama_, Feb 10 2017
%F a(n) = n*A013955(n) for n > 0. - _Seiichi Manyama_, Feb 18 2017
%F Sum_{k=1..n} a(k) ~ zeta(8) * n^9 / 9. - _Amiram Eldar_, Sep 06 2023
%F From _Amiram Eldar_, Oct 30 2023: (Start)
%F Multiplicative with a(p^e) = p^e * (p^(7*e+7)-1)/(p^7-1).
%F Dirichlet g.f.: zeta(s-1)*zeta(s-8). (End)
%e a(6) = 1^8*6^1 + 2^8*3^1 + 3^8*2^1 + 6^8*1^1 = 1693512.
%t terms = 25;
%t E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}];
%t E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
%t E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
%t E4[x]*(E2[x]*E4[x] - E6[x])/720 + O[x]^terms // CoefficientList[#, x]& (* _Jean-François Alcover_, Feb 26 2018 *)
%t Table[n * DivisorSigma[7, n], {n, 0, 24}] (* _Amiram Eldar_, Sep 06 2023 *)
%o (PARI) a(n) = if(n < 1, 0, n*sigma(n, 7)) \\ _Andrew Howroyd_, Jul 25 2018
%Y Cf. A064987 (phi_{2, 1}), A281372 (phi_{4, 1}), A282050 (phi_{6, 1}, this sequence (phi_{8, 1}).
%Y Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A282101 (E_2*E_4^2), A013974 (E_4*E_6 = E_10).
%Y Cf. A013955, A013666, A196288.
%K nonn,easy,mult
%O 0,3
%A _Seiichi Manyama_, Feb 05 2017
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